Inequalities for branching processes

C. R. Heathcote; E. Seneta

doi:10.2307/3212050

Inequalities for branching processes

Journal of Applied Probability ◽

10.1017/s0021900200114081 ◽

1966 ◽

Vol 3 (01) ◽

pp. 261-267 ◽

Cited By ~ 7

Author(s):

C. R. Heathcote ◽

E. Seneta

Keyword(s):

Generating Function ◽

Conditional Distribution ◽

Branching Process ◽

Branching Processes ◽

Probability Generating Function ◽

Random Variable ◽

Expected Time ◽

Time To Extinction ◽

Explicit Formulae

Summary If F(s) is the probability generating function of a non-negative random variable, the nth functional iterate Fn (s) = Fn– 1 (F(s)) generates the distribution of the size of the nth generation of a simple branching process. In general it is not possible to obtain explicit formulae for many quantities involving Fn (s), and this paper considers certain bounds and approximations. Bounds are found for the Koenigs-type iterates lim n→∞ m −n {1−Fn (s)}, 0 ≦ s ≦ 1 where m = F′ (1) < 1 and F′′ (1) < ∞; for the expected time to extinction and for the limiting conditional-distribution generating function limn→∞{Fn (s) − Fn (0)} [1 – Fn (0)]–1. Particular attention is paid to the case F(s) = exp {m(s − 1)}.

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Bounds on the extinction time distribution of a branching process

Advances in Applied Probability ◽

10.2307/1426296 ◽

1974 ◽

Vol 6 (2) ◽

pp. 322-335 ◽

Cited By ~ 25

Author(s):

Alan Agresti

Keyword(s):

Generating Function ◽

Generating Functions ◽

Time Distribution ◽

Branching Process ◽

Probability Generating Function ◽

Extinction Time ◽

Expected Time ◽

Time To Extinction ◽

Special Case ◽

Better Than

The class of fractional linear generating functions, one of the few known classes of probability generating functions whose iterates can be explicitly stated, is examined. The method of bounding a probability generating function g (satisfying g″(1) < ∞) by two fractional linear generating functions is used to derive bounds for the extinction time distribution of the Galton-Watson branching process with offspring probability distribution represented by g. For the special case of the Poisson probability generating function, the best possible bounding fractional linear generating functions are obtained, and the bounds for the expected time to extinction of the corresponding Poisson branching process are better than any previously published.

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Bounds on the extinction time distribution of a branching process

Advances in Applied Probability ◽

10.1017/s0001867800045390 ◽

1974 ◽

Vol 6 (02) ◽

pp. 322-335 ◽

Cited By ~ 13

Author(s):

Alan Agresti

Keyword(s):

Generating Function ◽

Generating Functions ◽

Time Distribution ◽

Branching Process ◽

Probability Generating Function ◽

Extinction Time ◽

Expected Time ◽

Time To Extinction ◽

Special Case ◽

Better Than

The class of fractional linear generating functions, one of the few known classes of probability generating functions whose iterates can be explicitly stated, is examined. The method of bounding a probability generating function g (satisfying g″(1) < ∞) by two fractional linear generating functions is used to derive bounds for the extinction time distribution of the Galton-Watson branching process with offspring probability distribution represented by g. For the special case of the Poisson probability generating function, the best possible bounding fractional linear generating functions are obtained, and the bounds for the expected time to extinction of the corresponding Poisson branching process are better than any previously published.

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A note on a functional equation arising in Galton-Watson branching processes

Journal of Applied Probability ◽

10.2307/3212181 ◽

1971 ◽

Vol 8 (3) ◽

pp. 589-598 ◽

Cited By ~ 18

Author(s):

Krishna B. Athreya

Keyword(s):

Functional Equation ◽

Continuous Function ◽

Generating Function ◽

Branching Process ◽

Branching Processes ◽

Probability Generating Function ◽

Age Dependent ◽

Watson Process ◽

Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

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A note on a functional equation arising in Galton-Watson branching processes

Journal of Applied Probability ◽

10.1017/s002190020003566x ◽

1971 ◽

Vol 8 (03) ◽

pp. 589-598 ◽

Cited By ~ 6

Author(s):

Krishna B. Athreya

Keyword(s):

Functional Equation ◽

Continuous Function ◽

Generating Function ◽

Branching Process ◽

Branching Processes ◽

Probability Generating Function ◽

Age Dependent ◽

Watson Process ◽

Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

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On expected durations of birth–death processes, with applications to branching processes and SIS epidemics

Journal of Applied Probability ◽

10.1017/jpr.2015.19 ◽

2016 ◽

Vol 53 (1) ◽

pp. 203-215 ◽

Cited By ~ 10

Author(s):

Frank Ball ◽

Tom Britton ◽

Peter Neal

Keyword(s):

Asymptotic Expression ◽

Branching Processes ◽

Random Variable ◽

Death Process ◽

Single Individual ◽

Expected Time ◽

Time To Extinction ◽

Number Of Individuals ◽

Sis Epidemic ◽

Birth Death

Abstract We study continuous-time birth–death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q] = 1, and where the birth rate if the population is currently in state (has size) n is α(n). We focus on two important examples, namely α(n) = λ n being a branching process, and α(n) = λn(N - n) / N which corresponds to an SIS (susceptible → infective → susceptible) epidemic model in a homogeneously mixing community of fixed size N. The processes are assumed to start with a single individual, i.e. in state 1. Let T, An, C, and S denote the (random) time to extinction, the total time spent in state n, the total number of individuals ever alive, and the sum of the lifetimes of all individuals in the birth–death process, respectively. We give expressions for the expectation of all these quantities and show that these expectations are insensitive to the distribution of Q. We also derive an asymptotic expression for the expected time to extinction of the SIS epidemic, but now starting at the endemic state, which is not independent of the distribution of Q. The results are also applied to the household SIS epidemic, showing that, in contrast to the household SIR (susceptible → infective → recovered) epidemic, its threshold parameter R* is insensitive to the distribution of Q.

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On asymptotic properties of sub-critical branching processes

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700006492 ◽

1968 ◽

Vol 8 (4) ◽

pp. 671-682 ◽

Cited By ~ 24

Author(s):

E. Seneta

Keyword(s):

Generating Function ◽

Branching Process ◽

Branching Processes ◽

Asymptotic Properties ◽

Probability Generating Function ◽

Image Position

Let Zn be the numer of individuals in the nth generation of a discrete branching process, descended from a single a singel ancestor, for which we put It is well known that the probability generating function of Zn is Fn(s), the n-th functional iterate of F(s), and that if m = EZ1 does not exceed unity, then lim (Harris [1], Chapter 1). In particular, extinction is certain.

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Decomposable branching processes with two types of particles

Discrete Mathematics and Applications ◽

10.1515/dma-2018-0012 ◽

2018 ◽

Vol 28 (2) ◽

pp. 119-130 ◽

Cited By ~ 2

Author(s):

Vladimir A. Vatutin ◽

Elena E. Dyakonova

Keyword(s):

Asymptotic Behavior ◽

Discrete Time ◽

Conditional Distribution ◽

Limit Theorems ◽

Branching Process ◽

Branching Processes ◽

Random Variable ◽

Infinite Variance ◽

Tail Distribution

Abstract A two-type critical decomposable branching process with discrete time is considered in which particles of the first type may produce at the death moment offspring of both types while particles of the second type may produce at the death moment offspring of their own type only. Assuming that the offspring distributions of particles of both types may have infinite variance, the asymptotic behavior of the tail distribution of the random variable Ξ2, the total number of the second type particles ever born in the process is found. Limit theorems are proved describing (as N → ∞) the conditional distribution of the amount of the first type particles in different generations given that either Ξ2 = N or Ξ2 > N.

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Period-Life of a Branching Process with Migration and Continuous Time

Mathematics ◽

10.3390/math9080868 ◽

2021 ◽

Vol 9 (8) ◽

pp. 868

Author(s):

Khrystyna Prysyazhnyk ◽

Iryna Bazylevych ◽

Ludmila Mitkova ◽

Iryna Ivanochko

Keyword(s):

Random Process ◽

Generating Function ◽

Continuous Time ◽

Branching Process ◽

Probability Generating Function ◽

Time Interval ◽

The Moment ◽

First Time ◽

Critical Branching Process ◽

Number Of Particles

The homogeneous branching process with migration and continuous time is considered. We investigated the distribution of the period-life τ, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time. The probability generating function of the random process, which describes the behavior of the process within the period-life, was obtained. The boundary theorem for the period-life of the subcritical or critical branching process with migration was found.

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Polynomial bounds for probability generating functions

Journal of Applied Probability ◽

10.2307/3212865 ◽

1975 ◽

Vol 12 (3) ◽

pp. 507-514 ◽

Cited By ~ 8

Author(s):

Henry Braun

Keyword(s):

Lower Bound ◽

Generating Function ◽

Generating Functions ◽

Branching Process ◽

Probability Generating Function ◽

Extinction Time ◽

Probability Generating Functions ◽

Extinction Probabilities ◽

Polynomial Bounds

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN(·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.

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